Rosa's TurbineBenchmark

Percentage Accurate: 85.6% → 98.1%

Time: 13.3s

Alternatives: 11

Speedup: 1.6×

• 53.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Initial Program: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Alternative 1: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 10^{+119}:\\ \;\;\;\;\left(3 + t\_0\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \mathsf{fma}\left(w, \left(w \cdot r\right) \cdot \left(r \cdot -0.375\right), -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 1e+119)
     (-
      (+ 3.0 t_0)
      (fma (* 0.125 (fma v -2.0 3.0)) (* (* w (* w r)) (/ r (- 1.0 v))) 4.5))
     (+ t_0 (fma w (* (* w r) (* r -0.375)) -1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 1e+119) {
		tmp = (3.0 + t_0) - fma((0.125 * fma(v, -2.0, 3.0)), ((w * (w * r)) * (r / (1.0 - v))), 4.5);
	} else {
		tmp = t_0 + fma(w, ((w * r) * (r * -0.375)), -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 1e+119)
		tmp = Float64(Float64(3.0 + t_0) - fma(Float64(0.125 * fma(v, -2.0, 3.0)), Float64(Float64(w * Float64(w * r)) * Float64(r / Float64(1.0 - v))), 4.5));
	else
		tmp = Float64(t_0 + fma(w, Float64(Float64(w * r) * Float64(r * -0.375)), -1.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 1e+119], N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(0.125 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(w * N[(N[(w * r), $MachinePrecision] * N[(r * -0.375), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 9.99999999999999944e118

   1. Initial program 89.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)} \]

   if 9.99999999999999944e118 < (*.f64 w w)

   1. Initial program 76.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 77.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + \left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot \frac{-3}{8}, \frac{-3}{2}\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot \frac{-3}{8}, \frac{-3}{2}\right) + \frac{2}{r \cdot r}} \]

      8. lower-+.f64 77.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right) + \frac{2}{r \cdot r}} \]

      9. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right)} + \frac{2}{r \cdot r} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      11. associate-*l* N/A

          \[\leadsto \left(\color{blue}{w \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)\right)} + \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(w, w \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)} + \frac{2}{r \cdot r} \]

      13. lower-*.f64 99.1

          \[\leadsto \mathsf{fma}\left(w, \color{blue}{w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right)}, -1.5\right) + \frac{2}{r \cdot r} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w, w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right), -1.5\right) + \frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(w, w \cdot \color{blue}{\left(r \cdot \left(r \cdot \frac{-3}{8}\right)\right)}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(w \cdot r\right) \cdot \left(r \cdot \frac{-3}{8}\right)}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot \frac{-3}{8}\right), \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot \frac{-3}{8}\right), \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot \frac{-3}{8}\right)}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      6. lower-*.f64 99.1

         \[\leadsto \mathsf{fma}\left(w, \left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot -0.375\right)}, -1.5\right) + \frac{2}{r \cdot r} \]

   9. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot -0.375\right)}, -1.5\right) + \frac{2}{r \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+119}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(w, \left(w \cdot r\right) \cdot \left(r \cdot -0.375\right), -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{v + -1} \leq 3:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right), -0.375, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* r (* (* w w) r))) (+ v -1.0)))
        3.0)
     (fma (* r (* w (* w r))) -0.375 -1.5)
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * ((w * w) * r))) / (v + -1.0))) <= 3.0) {
		tmp = fma((r * (w * (w * r))), -0.375, -1.5);
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r * Float64(Float64(w * w) * r))) / Float64(v + -1.0))) <= 3.0)
		tmp = fma(Float64(r * Float64(w * Float64(w * r))), -0.375, -1.5);
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], N[(N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + -1.5), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < 3

   1. Initial program 83.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around inf

      \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {w}^{2} \cdot \color{blue}{\left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{{w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + {w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left({w}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right) \cdot {w}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{w}^{2}} \cdot {w}^{2}\right)}\right)\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\frac{-3}{2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}, \frac{-3}{2}\right)} \]

   8. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \mathsf{fma}\left(r, r \cdot -0.375, \frac{2}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right), -1.5\right)} \]

   9. Taylor expanded in r around inf

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right) + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)} + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}} + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \color{blue}{\left(\mathsf{neg}\left({r}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)}\right)\right) \]

       8. lft-mult-inverse N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \color{blue}{\frac{-3}{2}} \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left({w}^{2} \cdot {r}^{2}\right)} \cdot \frac{-3}{8} + \frac{-3}{2} \]

       12. associate-*l* N/A

           \[\leadsto \color{blue}{{w}^{2} \cdot \left({r}^{2} \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       13. *-commutative N/A

           \[\leadsto {w}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right)} + \frac{-3}{2} \]

       14. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   11. Applied rewrites 73.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) + \frac{-3}{2} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       4. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right) \cdot \frac{-3}{8}} + \frac{-3}{2} \]

       6. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right), \frac{-3}{8}, \frac{-3}{2}\right)} \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right)} \cdot \left(w \cdot w\right), \frac{-3}{8}, \frac{-3}{2}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(r \cdot r\right) \cdot \color{blue}{\left(w \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       10. swap-sqr N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       14. lower-*.f64 87.1

           \[\leadsto \mathsf{fma}\left(r \cdot \color{blue}{\left(w \cdot \left(r \cdot w\right)\right)}, -0.375, -1.5\right) \]

   13. Applied rewrites 87.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right), -0.375, -1.5\right)} \]

   if 3 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 86.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. lower-*.f64 99.9

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{v + -1} \leq 3:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right), -0.375, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{v + -1} \leq 2.999999999999659:\\ \;\;\;\;\mathsf{fma}\left(r, \left(w \cdot w\right) \cdot \left(r \cdot -0.375\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* r (* (* w w) r))) (+ v -1.0)))
        2.999999999999659)
     (fma r (* (* w w) (* r -0.375)) -1.5)
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * ((w * w) * r))) / (v + -1.0))) <= 2.999999999999659) {
		tmp = fma(r, ((w * w) * (r * -0.375)), -1.5);
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r * Float64(Float64(w * w) * r))) / Float64(v + -1.0))) <= 2.999999999999659)
		tmp = fma(r, Float64(Float64(w * w) * Float64(r * -0.375)), -1.5);
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.999999999999659], N[(r * N[(N[(w * w), $MachinePrecision] * N[(r * -0.375), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < 2.999999999999659

   1. Initial program 87.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around inf

      \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {w}^{2} \cdot \color{blue}{\left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{{w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + {w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left({w}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right) \cdot {w}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{w}^{2}} \cdot {w}^{2}\right)}\right)\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\frac{-3}{2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}, \frac{-3}{2}\right)} \]

   8. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \mathsf{fma}\left(r, r \cdot -0.375, \frac{2}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right), -1.5\right)} \]

   9. Taylor expanded in r around inf

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right) + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)} + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}} + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \color{blue}{\left(\mathsf{neg}\left({r}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)}\right)\right) \]

       8. lft-mult-inverse N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \color{blue}{\frac{-3}{2}} \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left({w}^{2} \cdot {r}^{2}\right)} \cdot \frac{-3}{8} + \frac{-3}{2} \]

       12. associate-*l* N/A

           \[\leadsto \color{blue}{{w}^{2} \cdot \left({r}^{2} \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       13. *-commutative N/A

           \[\leadsto {w}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right)} + \frac{-3}{2} \]

       14. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   11. Applied rewrites 81.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) + \frac{-3}{2} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) \cdot \left(w \cdot w\right)} + \frac{-3}{2} \]

       5. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} \cdot \left(w \cdot w\right) + \frac{-3}{2} \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) \cdot \left(w \cdot w\right) + \frac{-3}{2} \]

       7. associate-*l* N/A

          \[\leadsto \color{blue}{\left(r \cdot \left(r \cdot \frac{-3}{8}\right)\right)} \cdot \left(w \cdot w\right) + \frac{-3}{2} \]

       8. associate-*l* N/A

          \[\leadsto \color{blue}{r \cdot \left(\left(r \cdot \frac{-3}{8}\right) \cdot \left(w \cdot w\right)\right)} + \frac{-3}{2} \]

       9. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(r, \left(r \cdot \frac{-3}{8}\right) \cdot \left(w \cdot w\right), \frac{-3}{2}\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(r, \color{blue}{\left(r \cdot \frac{-3}{8}\right) \cdot \left(w \cdot w\right)}, \frac{-3}{2}\right) \]

       11. lower-*.f64 86.0

           \[\leadsto \mathsf{fma}\left(r, \color{blue}{\left(r \cdot -0.375\right)} \cdot \left(w \cdot w\right), -1.5\right) \]

   13. Applied rewrites 86.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(r, \left(r \cdot -0.375\right) \cdot \left(w \cdot w\right), -1.5\right)} \]

   if 2.999999999999659 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 83.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. lower-*.f64 93.6

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{v + -1} \leq 2.999999999999659:\\ \;\;\;\;\mathsf{fma}\left(r, \left(w \cdot w\right) \cdot \left(r \cdot -0.375\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{v + -1} \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (+
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* r (* (* w w) r))) (+ v -1.0)))
        -2e+57)
     (* (* r r) (* w (* w -0.375)))
     (+ t_0 -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * ((w * w) * r))) / (v + -1.0))) <= -2e+57) {
		tmp = (r * r) * (w * (w * -0.375));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (((3.0d0 + t_0) + (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (r * ((w * w) * r))) / (v + (-1.0d0)))) <= (-2d+57)) then
        tmp = (r * r) * (w * (w * (-0.375d0)))
    else
        tmp = t_0 + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * ((w * w) * r))) / (v + -1.0))) <= -2e+57) {
		tmp = (r * r) * (w * (w * -0.375));
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * ((w * w) * r))) / (v + -1.0))) <= -2e+57:
		tmp = (r * r) * (w * (w * -0.375))
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r * Float64(Float64(w * w) * r))) / Float64(v + -1.0))) <= -2e+57)
		tmp = Float64(Float64(r * r) * Float64(w * Float64(w * -0.375)));
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r * ((w * w) * r))) / (v + -1.0))) <= -2e+57)
		tmp = (r * r) * (w * (w * -0.375));
	else
		tmp = t_0 + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+57], N[(N[(r * r), $MachinePrecision] * N[(w * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -2.0000000000000001e57

   1. Initial program 87.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 81.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around inf

      \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {w}^{2} \cdot \color{blue}{\left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{{w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + {w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left({w}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right) \cdot {w}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{w}^{2}} \cdot {w}^{2}\right)}\right)\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\frac{-3}{2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}, \frac{-3}{2}\right)} \]

   8. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \mathsf{fma}\left(r, r \cdot -0.375, \frac{2}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right), -1.5\right)} \]

   9. Taylor expanded in w around inf

      \[\leadsto \color{blue}{\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{{r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)} \]

       3. *-commutative N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2}\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right) \]

       7. *-commutative N/A

          \[\leadsto \left(r \cdot r\right) \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)} \]

       8. unpow2 N/A

          \[\leadsto \left(r \cdot r\right) \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot \frac{-3}{8}\right) \]

       9. associate-*l* N/A

          \[\leadsto \left(r \cdot r\right) \cdot \color{blue}{\left(w \cdot \left(w \cdot \frac{-3}{8}\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(r \cdot r\right) \cdot \color{blue}{\left(w \cdot \left(w \cdot \frac{-3}{8}\right)\right)} \]

       11. lower-*.f64 81.2

           \[\leadsto \left(r \cdot r\right) \cdot \left(w \cdot \color{blue}{\left(w \cdot -0.375\right)}\right) \]

   11. Applied rewrites 81.2%

       \[\leadsto \color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot -0.375\right)\right)} \]

   if -2.0000000000000001e57 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 83.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. lower-*.f64 93.4

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Applied rewrites 93.4%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)}{v + -1} \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 10000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(w, \left(w \cdot r\right) \cdot \left(r \cdot -0.375\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 10000000.0)
   (+ (/ 2.0 (* r r)) (fma w (* (* w r) (* r -0.375)) -1.5))
   (-
    3.0
    (fma (* 0.125 (fma v -2.0 3.0)) (* (* w (* w r)) (/ r (- 1.0 v))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 10000000.0) {
		tmp = (2.0 / (r * r)) + fma(w, ((w * r) * (r * -0.375)), -1.5);
	} else {
		tmp = 3.0 - fma((0.125 * fma(v, -2.0, 3.0)), ((w * (w * r)) * (r / (1.0 - v))), 4.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 10000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + fma(w, Float64(Float64(w * r) * Float64(r * -0.375)), -1.5));
	else
		tmp = Float64(3.0 - fma(Float64(0.125 * fma(v, -2.0, 3.0)), Float64(Float64(w * Float64(w * r)) * Float64(r / Float64(1.0 - v))), 4.5));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 10000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(w * N[(N[(w * r), $MachinePrecision] * N[(r * -0.375), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(0.125 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1e7

   1. Initial program 85.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + \left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot \frac{-3}{8}, \frac{-3}{2}\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot \frac{-3}{8}, \frac{-3}{2}\right) + \frac{2}{r \cdot r}} \]

      8. lower-+.f64 79.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right) + \frac{2}{r \cdot r}} \]

      9. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right)} + \frac{2}{r \cdot r} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      11. associate-*l* N/A

          \[\leadsto \left(\color{blue}{w \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)\right)} + \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(w, w \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)} + \frac{2}{r \cdot r} \]

      13. lower-*.f64 89.8

          \[\leadsto \mathsf{fma}\left(w, \color{blue}{w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right)}, -1.5\right) + \frac{2}{r \cdot r} \]

   7. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w, w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right), -1.5\right) + \frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(w, w \cdot \color{blue}{\left(r \cdot \left(r \cdot \frac{-3}{8}\right)\right)}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(w \cdot r\right) \cdot \left(r \cdot \frac{-3}{8}\right)}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot \frac{-3}{8}\right), \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot \frac{-3}{8}\right), \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot \frac{-3}{8}\right)}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      6. lower-*.f64 94.8

         \[\leadsto \mathsf{fma}\left(w, \left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot -0.375\right)}, -1.5\right) + \frac{2}{r \cdot r} \]

   9. Applied rewrites 94.8%

      \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot -0.375\right)}, -1.5\right) + \frac{2}{r \cdot r} \]

   if 1e7 < r

   1. Initial program 84.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)} \]

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, \frac{9}{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 98.3%

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 10000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(w, \left(w \cdot r\right) \cdot \left(r \cdot -0.375\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 10^{+49}:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(w, \left(w \cdot r\right) \cdot \left(r \cdot -0.375\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right), -0.375, -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1e+49)
   (+ (/ 2.0 (* r r)) (fma w (* (* w r) (* r -0.375)) -1.5))
   (fma (* r (* w (* w r))) -0.375 -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1e+49) {
		tmp = (2.0 / (r * r)) + fma(w, ((w * r) * (r * -0.375)), -1.5);
	} else {
		tmp = fma((r * (w * (w * r))), -0.375, -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1e+49)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + fma(w, Float64(Float64(w * r) * Float64(r * -0.375)), -1.5));
	else
		tmp = fma(Float64(r * Float64(w * Float64(w * r))), -0.375, -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1e+49], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(w * N[(N[(w * r), $MachinePrecision] * N[(r * -0.375), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 9.99999999999999946e48

   1. Initial program 85.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + \left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot \frac{-3}{8}, \frac{-3}{2}\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot \frac{-3}{8}, \frac{-3}{2}\right) + \frac{2}{r \cdot r}} \]

      8. lower-+.f64 80.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right) + \frac{2}{r \cdot r}} \]

      9. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right)} + \frac{2}{r \cdot r} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      11. associate-*l* N/A

          \[\leadsto \left(\color{blue}{w \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)\right)} + \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(w, w \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)} + \frac{2}{r \cdot r} \]

      13. lower-*.f64 89.9

          \[\leadsto \mathsf{fma}\left(w, \color{blue}{w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right)}, -1.5\right) + \frac{2}{r \cdot r} \]

   7. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w, w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right), -1.5\right) + \frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(w, w \cdot \color{blue}{\left(r \cdot \left(r \cdot \frac{-3}{8}\right)\right)}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(w \cdot r\right) \cdot \left(r \cdot \frac{-3}{8}\right)}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot \frac{-3}{8}\right), \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right)} \cdot \left(r \cdot \frac{-3}{8}\right), \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot \frac{-3}{8}\right)}, \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      6. lower-*.f64 94.6

         \[\leadsto \mathsf{fma}\left(w, \left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot -0.375\right)}, -1.5\right) + \frac{2}{r \cdot r} \]

   9. Applied rewrites 94.6%

      \[\leadsto \mathsf{fma}\left(w, \color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot -0.375\right)}, -1.5\right) + \frac{2}{r \cdot r} \]

   if 9.99999999999999946e48 < r

   1. Initial program 83.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around inf

      \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {w}^{2} \cdot \color{blue}{\left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{{w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + {w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left({w}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right) \cdot {w}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{w}^{2}} \cdot {w}^{2}\right)}\right)\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\frac{-3}{2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}, \frac{-3}{2}\right)} \]

   8. Applied rewrites 67.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \mathsf{fma}\left(r, r \cdot -0.375, \frac{2}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right), -1.5\right)} \]

   9. Taylor expanded in r around inf

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right) + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)} + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}} + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \color{blue}{\left(\mathsf{neg}\left({r}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)}\right)\right) \]

       8. lft-mult-inverse N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \color{blue}{\frac{-3}{2}} \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left({w}^{2} \cdot {r}^{2}\right)} \cdot \frac{-3}{8} + \frac{-3}{2} \]

       12. associate-*l* N/A

           \[\leadsto \color{blue}{{w}^{2} \cdot \left({r}^{2} \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       13. *-commutative N/A

           \[\leadsto {w}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right)} + \frac{-3}{2} \]

       14. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   11. Applied rewrites 76.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) + \frac{-3}{2} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       4. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right) \cdot \frac{-3}{8}} + \frac{-3}{2} \]

       6. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right), \frac{-3}{8}, \frac{-3}{2}\right)} \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right)} \cdot \left(w \cdot w\right), \frac{-3}{8}, \frac{-3}{2}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(r \cdot r\right) \cdot \color{blue}{\left(w \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       10. swap-sqr N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       14. lower-*.f64 87.3

           \[\leadsto \mathsf{fma}\left(r \cdot \color{blue}{\left(w \cdot \left(r \cdot w\right)\right)}, -0.375, -1.5\right) \]

   13. Applied rewrites 87.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right), -0.375, -1.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 10^{+49}:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(w, \left(w \cdot r\right) \cdot \left(r \cdot -0.375\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right), -0.375, -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(w, w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right), -0.375, -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5e+149)
   (+ (/ 2.0 (* r r)) (fma w (* w (* (* r r) -0.375)) -1.5))
   (fma (* r (* w (* w r))) -0.375 -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e+149) {
		tmp = (2.0 / (r * r)) + fma(w, (w * ((r * r) * -0.375)), -1.5);
	} else {
		tmp = fma((r * (w * (w * r))), -0.375, -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5e+149)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + fma(w, Float64(w * Float64(Float64(r * r) * -0.375)), -1.5));
	else
		tmp = fma(Float64(r * Float64(w * Float64(w * r))), -0.375, -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5e+149], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(w * N[(w * N[(N[(r * r), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.9999999999999999e149

   1. Initial program 85.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + \left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot \frac{-3}{8}, \frac{-3}{2}\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot \frac{-3}{8}, \frac{-3}{2}\right) + \frac{2}{r \cdot r}} \]

      8. lower-+.f64 80.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right) + \frac{2}{r \cdot r}} \]

      9. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right)} + \frac{2}{r \cdot r} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      11. associate-*l* N/A

          \[\leadsto \left(\color{blue}{w \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)\right)} + \frac{-3}{2}\right) + \frac{2}{r \cdot r} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(w, w \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)} + \frac{2}{r \cdot r} \]

      13. lower-*.f64 89.6

          \[\leadsto \mathsf{fma}\left(w, \color{blue}{w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right)}, -1.5\right) + \frac{2}{r \cdot r} \]

   7. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w, w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right), -1.5\right) + \frac{2}{r \cdot r}} \]

   if 4.9999999999999999e149 < r

   1. Initial program 79.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around inf

      \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {w}^{2} \cdot \color{blue}{\left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{{w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + {w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left({w}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right) \cdot {w}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{w}^{2}} \cdot {w}^{2}\right)}\right)\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\frac{-3}{2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}, \frac{-3}{2}\right)} \]

   8. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \mathsf{fma}\left(r, r \cdot -0.375, \frac{2}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right), -1.5\right)} \]

   9. Taylor expanded in r around inf

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right) + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)} + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}} + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \color{blue}{\left(\mathsf{neg}\left({r}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)}\right)\right) \]

       8. lft-mult-inverse N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \color{blue}{\frac{-3}{2}} \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left({w}^{2} \cdot {r}^{2}\right)} \cdot \frac{-3}{8} + \frac{-3}{2} \]

       12. associate-*l* N/A

           \[\leadsto \color{blue}{{w}^{2} \cdot \left({r}^{2} \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       13. *-commutative N/A

           \[\leadsto {w}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right)} + \frac{-3}{2} \]

       14. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   11. Applied rewrites 70.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) + \frac{-3}{2} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       4. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right) \cdot \frac{-3}{8}} + \frac{-3}{2} \]

       6. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right), \frac{-3}{8}, \frac{-3}{2}\right)} \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right)} \cdot \left(w \cdot w\right), \frac{-3}{8}, \frac{-3}{2}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(r \cdot r\right) \cdot \color{blue}{\left(w \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       10. swap-sqr N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       14. lower-*.f64 88.1

           \[\leadsto \mathsf{fma}\left(r \cdot \color{blue}{\left(w \cdot \left(r \cdot w\right)\right)}, -0.375, -1.5\right) \]

   13. Applied rewrites 88.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right), -0.375, -1.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(w, w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right), -0.375, -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 14.2:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right), -0.375, -1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 14.2)
   (+ -1.5 (fma (* w (* (* r r) -0.25)) w (/ 2.0 (* r r))))
   (fma (* r (* w (* w r))) -0.375 -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 14.2) {
		tmp = -1.5 + fma((w * ((r * r) * -0.25)), w, (2.0 / (r * r)));
	} else {
		tmp = fma((r * (w * (w * r))), -0.375, -1.5);
	}
	return tmp;
}

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 14.2)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r * r) * -0.25)), w, Float64(2.0 / Float64(r * r))));
	else
		tmp = fma(Float64(r * Float64(w * Float64(w * r))), -0.375, -1.5);
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 14.2], N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 14.199999999999999

   1. Initial program 85.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Applied rewrites 87.0%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]

   if 14.199999999999999 < r

   1. Initial program 84.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around inf

      \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {w}^{2} \cdot \color{blue}{\left(\left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{{w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + {w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)} \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left({w}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right) \cdot {w}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{w}^{2}} \cdot {w}^{2}\right)}\right)\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto {w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}\right) + \color{blue}{\frac{-3}{2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2} + \frac{2}{{r}^{2} \cdot {w}^{2}}, \frac{-3}{2}\right)} \]

   8. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \mathsf{fma}\left(r, r \cdot -0.375, \frac{2}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right), -1.5\right)} \]

   9. Taylor expanded in r around inf

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right) + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)} + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}} + {r}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \color{blue}{\left(\mathsf{neg}\left({r}^{2} \cdot \left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \cdot {r}^{2}}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \left(\frac{1}{{r}^{2}} \cdot {r}^{2}\right)}\right)\right) \]

       8. lft-mult-inverse N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \color{blue}{1}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8} + \color{blue}{\frac{-3}{2}} \]

       11. *-commutative N/A

           \[\leadsto \color{blue}{\left({w}^{2} \cdot {r}^{2}\right)} \cdot \frac{-3}{8} + \frac{-3}{2} \]

       12. associate-*l* N/A

           \[\leadsto \color{blue}{{w}^{2} \cdot \left({r}^{2} \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       13. *-commutative N/A

           \[\leadsto {w}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {r}^{2}\right)} + \frac{-3}{2} \]

       14. lower-fma.f64 N/A

           \[\leadsto \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   11. Applied rewrites 79.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(w \cdot w\right)} \cdot \left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right) + \frac{-3}{2} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot \frac{-3}{8}\right) + \frac{-3}{2} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       4. lift-*.f64 N/A

          \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \frac{-3}{8}\right)} + \frac{-3}{2} \]

       5. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right) \cdot \frac{-3}{8}} + \frac{-3}{2} \]

       6. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right), \frac{-3}{8}, \frac{-3}{2}\right)} \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot r\right)} \cdot \left(w \cdot w\right), \frac{-3}{8}, \frac{-3}{2}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(r \cdot r\right) \cdot \color{blue}{\left(w \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       10. swap-sqr N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}, \frac{-3}{8}, \frac{-3}{2}\right) \]

       14. lower-*.f64 86.4

           \[\leadsto \mathsf{fma}\left(r \cdot \color{blue}{\left(w \cdot \left(r \cdot w\right)\right)}, -0.375, -1.5\right) \]

   13. Applied rewrites 86.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right), -0.375, -1.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 14.2:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right), -0.375, -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.15d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.15)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 85.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

      3. lower-*.f64 59.6

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

   5. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]

   if 1.1499999999999999 < r

   1. Initial program 84.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around 0

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\frac{-3}{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.8%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   9. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{-3}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 24.6%

       \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 56.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (+ (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (2.0d0 / (r * r)) + (-1.5d0)
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + -1.5;
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + -1.5

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + -1.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + -1.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]

Derivation

1. Initial program 85.0%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Add Preprocessing

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

   9. lower-*.f64 60.4

      \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

5. Applied rewrites 60.4%

   \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]

6. Final simplification 60.4%

   \[\leadsto \frac{2}{r \cdot r} + -1.5 \]
7. Add Preprocessing

Alternative 11: 14.6% accurate, 73.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return -1.5;
}

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 85.0%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

   9. distribute-neg-in N/A

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

   10. associate-*r* N/A

       \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

   12. distribute-rgt-neg-in N/A

       \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

5. Applied rewrites 79.7%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

6. Taylor expanded in w around 0

   \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\frac{-3}{2}} \]
7. Step-by-step derivation

8. Applied rewrites 60.4%

   \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

9. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{-3}{2}} \]
10. Step-by-step derivation

11. Applied rewrites 14.0%

    \[\leadsto \color{blue}{-1.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))